Madhava's correction term

Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π. The Madhava–Leibniz infinite series for π is π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots } Taking the partial sum of the first n {\displaystyle n} terms we have the following approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}} Denoting the Madhava correction term by F ( n ) {\displaystyle F(n)} , we have the following better approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F ( n ) {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F(n)} Three different expressions have been attributed to Madhava as possible values of F ( n ) {\displaystyle F(n)} , namely, F 1 ( n ) = 1 4 n {\displaystyle F_{1}(n)={\frac {1}{4n}}} F 2 ( n ) = n 4 n 2 + 1 {\displaystyle F_{2}(n)={\frac {n}{4n^{2}+1}}} F 3 ( n ) = n 2 + 1 4 n 3 + 5 n {\displaystyle F_{3}(n)={\frac {n^{2}+1}{4n^{3}+5n}}} In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms F 1 ( n ) {\displaystyle F_{1}(n)} and F 2 ( n ) {\displaystyle F_{2}(n)} have been obtained, but there are no indications on how the expression F 3 ( n ) {\displaystyle F_{3}(n)} has been obtained.

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Madhava's correction term

Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π. The Madhava–Leibniz infinite series for π is π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots } Taking the partial sum of the first n {\displaystyle n} terms we have the following approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}} Denoting the Madhava correction term by F ( n ) {\displaystyle F(n)} , we have the following better approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F ( n ) {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F(n)} Three different expressions have been attributed to Madhava as possible values of F ( n ) {\displaystyle F(n)} , namely, F 1 ( n ) = 1 4 n {\displaystyle F_{1}(n)={\frac {1}{4n}}} F 2 ( n ) = n 4 n 2 + 1 {\displaystyle F_{2}(n)={\frac {n}{4n^{2}+1}}} F 3 ( n ) = n 2 + 1 4 n 3 + 5 n {\displaystyle F_{3}(n)={\frac {n^{2}+1}{4n^{3}+5n}}} In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms F 1 ( n ) {\displaystyle F_{1}(n)} and F 2 ( n ) {\displaystyle F_{2}(n)} have been obtained, but there are no indications on how the expression F 3 ( n ) {\displaystyle F_{3}(n)} has been obtained.

Source: Wikipedia "Madhava's correction term" · CC BY-SA 4.0

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